Abstract
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.
Highlights
We can distinguish three categories of papers: 1) some early results based mostly on comparison methods: see [42, 37, 1, 41] and references therein; 2) a linearization motivated by the gradient flow structure of the fast diffusion equations: [22, 23, 24, 38]; 3) entropy based approaches: [4, 5, 7, 9, 11, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 39]
The parameter α is a measure of the intensity of the derivative in the radial direction compared to angular derivatives and plays a crucial role in the symmetry breaking issues, as shown by Condition (17)
As for symmetry breaking, what matters is to compare Λ and Λ : if Λ < Λ, by considering an eigenfunction associated with the spectral gap, we shall prove that Q takes negative eigenvalues, and this is what establishes the result of Theorem 2
Summary
D shrinks to the simple half-line given by β d−2 d γ, while the whole range of is covered in the limit as p → 1. Since Barenblatt type profiles attract all solutions at least when m ∈ [m1, 1), the linearization around these profiles is again enough to get an answer This is the purpose of our second main result. The condition (10) may look rather restrictive, but it is probably not, because it is expected that the condition is satisfied, for some positive t, by any solution with initial datum as in Proposition 1. At least this is what occurs when (β, γ) = (0, 0): see for instance [10, 5].
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